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Theorem 00lss 15981
Description: The empty structure has no subspaces (for use with fvco4i 5768). (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
00lss  |-  (/)  =  (
LSubSp `  (/) )

Proof of Theorem 00lss
StepHypRef Expression
1 noel 3600 . . 3  |-  -.  a  e.  (/)
2 base0 13469 . . . . . 6  |-  (/)  =  (
Base `  (/) )
3 eqid 2412 . . . . . 6  |-  ( LSubSp `  (/) )  =  ( LSubSp `
 (/) )
42, 3lssss 15976 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  C_  (/) )
5 ss0 3626 . . . . 5  |-  ( a 
C_  (/)  ->  a  =  (/) )
64, 5syl 16 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =  (/) )
73lssn0 15980 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =/=  (/) )
87neneqd 2591 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  ->  -.  a  =  (/) )
96, 8pm2.65i 167 . . 3  |-  -.  a  e.  ( LSubSp `  (/) )
101, 92false 340 . 2  |-  ( a  e.  (/)  <->  a  e.  (
LSubSp `  (/) ) )
1110eqriv 2409 1  |-  (/)  =  (
LSubSp `  (/) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721    C_ wss 3288   (/)c0 3596   ` cfv 5421   LSubSpclss 15971
This theorem is referenced by:  00lsp  16020  lidlval  16228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-slot 13436  df-base 13437  df-lss 15972
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